This paper is a sequel to arXiv:0902.2355 and continues the study of quantum logic via dagger kernel categories. It develops the relation between these categories and both orthomodular lattices and Foulis semigroups. The relation between the latter two notions has been uncovered in the 1960s. The current categorical perspective gives a broader context and reconstructs this relationship between orthomodular lattices and Foulis semigroups as special instance.

• 03G12 - Quantum logic
• 03G30 - Categorical logic, topoi
• 06C15 - Complemented lattices, orthocomplemented lattices and posets